manhattan review

nickhar130 wrote:How many integers between 250 and 300, inclusiv, can be evenly divided by NEITHER 3 NOR 5?

a) 33
b) 28'
c) 27
d) 25
e) 24

is there a trick to this, please advise?

Yes!!! Of course there is a trick.

What is the first multiple of 3 between 250 and 300? Its 252.
What is the last multiple of 3 between 250 and 300? Its 300.

How many are divisible by 3 between 252 and 300?
(300-252)/3 = 48/3 = 16 but it excludes 252. Hence 17.


What is the first multiple of 5 between 250 and 300? Its 250.
What is the last multiple of 5 between 250 and 300? Its 300.

How many are divisible by 3 between 252 and 300?
(300-250)/5 = 50/5 = 10 but it excludes 250. Hence 11.

But wait there are some integers which are multiple of 3 and 5. Thus we need to discard the integers which are multiple of LCM[3,5] = 15.

What is the first multiple of 15 between 250 and 300? Its 255.
What is the last multiple of 15 between 250 and 300? Its 300.

How many are divisible by 15 between 255 and 300?
(300-255)/15 = 45/15 = 3 but it excludes 255. Hence 4.

So, Total = 11 +17 -4 = 24

Total numbers between 250 and 300 is 300-250 + 1 = 51

Not divisible by 3 and 5 = 51 - 24 = 27

thank you all. Explanation is very clear.